Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians

Abstract

Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by using the inherent fundamental solution and the implicit variational principle of Hamilton dynamics with their Hamiltonian depending on the unknown. Secondly, under additional growth assumptions on the Hamiltonian, we establish global H\"older regularity for both the solutions and the correctors, serving as a notable application of our quantitative homogenization theory.